This paper proposes a new Hybrid Particle Swarm Optimization (HPSO) method that integrates the Evolutionary Programming (EP) and Particle Swarm Optimization (PSO) techniques. The proposed method is applied to solve Economic Dispatch(ED) problems considering prohibited operating zones, ramp rate limits, capacity limits and power balance constraints. In the proposed HPSO method, the best features of both EP and PSO are exploited, and it is capable of finding the most optimal solution for the nonlinear optimization problems. For validating the proposed method, it has been tested on the standard three, six, fifteen and twenty unit test systems. The numerical results show that the proposed HPSO method is well suitable for solving nonlinear economic dispatch problems, and it outperforms the EP, PSO and other modern metaheuristic optimization methods reported in the recent literatures.
1. Introduction
Increasing daytoday power demands, scarcity of energy resources and increasing power generation costs necessitates optimal economic dispatch(ED) in today’s power system. Economic dispatch problem has become one of the most important power system optimization problems in real time application.
The main objective of the economic dispatch problem in the power system is to find the optimal combination of power generation that minimizes the total fuel cost while satisfying the system constraints
[1]
. Many conventional methods such as Lambda iteration method, Newton’s method, Gradient method, Linear programming method, Interior point method and Dynamic programming method have been applied to solve the basic economic dispatch (ED) problems
[2]
. In all these methods, the fuel cost function considered as quadratic in nature. However, in reality, the inputoutput characteristics of the generating units are to be nonlinear due to prohibited operating zones, and ramp rate limit constraints. The Lambdaiteration method has been applied to many software packages and used by power utilities for solving ED problems due to ease of implementation.
Since the lambda iteration method requires a continuous problem formulation, it cannot be directly applied to ED problems with discontinuous prohibited operating zones. For the selection of initial conditions, Newton’s method is very much sensitive
[3]
. Dynamic Programming (DP) method is one of the best conventional approach to solve the ED problems with nonconvex and unit cost functions. However, the DP method may cause the problems of the curse of dimensionality or local optimality
[4]
in the solution procedure
Practically, ED problem is nonlinear, nonconvex type with multiple local optimal points due to inclusion of equality, inequality constraints, and prohibited operating zones. Conventional methods have failed to solve such type of problems and converge into local optimal solution
[5]
. All these methods assume that the cost curve is continuous and monotonically increasing. To overcome the problems of conventional methods for solving ED problems, the researcher’s have put into their step by using modern metaheuristic searching techniques, including Simulated Annealing (SA)
[6]
, Modified Hopfield Network method
[7]
, Genetic Algorithm method (GA)
[8]
, Evolutionary Programming method
[9

13]
, Tabu Search algorithm (TSA)
[14]
, Particle Swarm Optimization method (PSO)
[15

18]
have been applied to solve the complex nonlinear ED problems. But these methods do not always guarantee a global optimal solution.
In Simulated Annealing method, Annealing schedule is very closely related to performance optimization. However, a poor tuning of the annealing schedule may inadvertently affect the performance of simulated annealing. Hop field neural network method requires external training routines. Recent researchers have identified some deficiencies in GA performance
[8]
. The premature convergence of GA degrades its performance and reduces its search capability that leads to a higher probability towards obtaining only the local optimal solutions
[15]
. The another drawback of GA is premature convergence leading to local minima and the complicated process in coding and decoding the problem
[19]
. Evolutionary Programming method for ED problem is more efficient than GA method in computation time and can generate a highquality solution with a shorter calculation. Particle swarm optimization is one of the latest versions of nature inspired algorithms which characteristics of high performance and easy implementation. PSO has a character of parallel searching mechanism, so it has high probability to determine the global (or) near global optimal solutions for the nonlinear ED problems. The main drawback of the conventional PSO is its premature convergence, especially while handling the problems with more local optima and heavier constraints
[19]
. The another drawback of PSO is sensitive to the tuning of some parameters and weighting factors. The proper and appropriate parameter tuning is absolutely necessary for quality solution. In order to overcome this troublesome parameter setting process, many researchers have proposed adaptive techniques. Zong Woo Geem has proposed parameter setting free Harmony search (PSFHS) technique to solve economic dispatch problem
[20]
. The results of PSFHS technique are quite encouraging in terms of convergence pattern and solution quality.
Various attempts have been made to overcome the problem of conventional (normal) PSO. Adaptive optimization algorithm must obtain a better balance between the local and global search ability, which means that the algorithm must has the ability to maintain a better local exploitation and global exploration ability. Among them, many adaptive approaches and strategies are proposed to enhance the performance of PSO. Self adaptive real coded GA
[21]
, Iteration PSO with time varying acceleration coefficient
[22]
have been proposed to solve different types of nonconvex ED problems. One of the wellknown improved PSO algorithms of the parameter modifying method is inertia weight PSO, by introducing the inertia weight; the performance of the conventional PSO is improved. Empirical studies of PSO with inertia weight have been shown that a relatively large value of w have more global search ability while a relatively small value of w results in a faster convergence.
The performance of the PSO via adjusting inertia weight such as Fuzzy adaptive particle swarm optimization
[23]
Linearly Decreasing Weight [LDW]
[24]
Increasing Inertia Weight
[25]
and Randomized Inertia Weight
[26
,
27]
have been proposed to solve different types of ED problems. In
[24]
, Shi and Eberhart introduced the inertia weight to the velocity update equation of the original PSO. The present of the inertia weight increases the convergence speed greatly, and obtain a better balance between exploitation and exploration of the solution space while having little increase of the algorithm complexity. The strategy of linearly decreasing weight (LDW) is most commonly used and it can improve the performance of PSO to some extent, but it may be trapped in local optima and fail to attain high search accuracy.
In recent years, combinations of two different optimization techniques were introduced by researcher’s to improve their earlier results. The following quoted here are some examples from recent literatures, which have used the combination of two different optimization techniques to solve the non linear economic dispatch problems. Simulated AnnealingParticle Swarm Optimization (SAPSO)
[28]
, Self Tuning Hybrid Differential Evolution (STH DE)
[29]
, Variable Scaling Hybrid Differential Evolution (VSHDE)
[30]
, Improved Genetic Algorithm with Multiplier Updating (IGAMU)
[31]
, Quantuminspired version of the PSO using the harmonic oscillator (HQPSO)
[32]
, SelfOrganizing hierarchical Particle Swarm Optimization (SOHPSO)
[33]
, and Bacterial Forging with NelderMead Algorithm (BFANM)
[34]
.
The main objective of the present work is to develop a hybrid algorithm which will be suitable for larger systems and to avoid premature convergence. The result obtained by the proposed algorithm is compared with EP, PSO which are developed using MATLAB and also with other intelligent techniques reported in the recent literatures.
The rest of this paper is organized as follows : section 2 introduces the problem formulation; section 3 explains over view of EP and PSO; section 4 presents a description of step by step development and solution methodology of the proposed HPSO method; section 5 shows the results and discussion and conclusion is summarized in section 6.
2. Problem Formulation
The objective of ED problem is to minimize the total generation cost of thermal generating units while satisfying various system constraints, including power balance equation, generator power limits, prohibited operating zones and ramp rate limit constraints.
The problem of ED is multimodal, nondifferentiable and highly nonlinear. Mathematically, the problem can be stated as in (1)
[2
,
21]
where F
_{T}
is the total fuel cost, N is the number of generating units in the system.
F_{i}
(
P_{i}
) is the fuel cost function of unit i and P
_{i}
is the output power of unit i. Generally, the fuel cost of generation unit can be expressed as
Where a
_{i}
, b
_{i}
and c
_{i}
are the cost coefficients of unit i subjected to the following constraints.
 2.1 Real power balance constraint
where
P_{D}
is real power demand and
P_{L}
is the transmission loss.
The transmission loss (
P_{L}
) can be expressed in a quadratic function of generation (Using Bloss coefficient matrix).
where
P_{i}
and
P_{j}
are the power generation of i
^{th}
and j
^{th}
units and
B_{ij}
,
B
_{0i}
, B
_{00}
are the B – loss coefficients.
 2.2 Generator operating limits
The power output of each unit i is restricted by its maximum and minimum limits of real power generation and is given by
where
P
_{i min}
and
P
_{i max}
are the minimum and maximum generation limits on i
^{th}
unit respectively.
 2.3 Prohibited operating zone constraints
The generators may have the certain range where operation is restricted due to the physical limitation of steam valve, component, vibration in shaft bearing etc., The consideration of prohibited operating zone (poz) creates a discontinuity in fuel cost curve and converts the constraint as below
where,
P^{L}_{i,k}
and
P^{u}_{i,k}
are the lower and upper boundary of k
^{th}
prohibited operating zone of unit i, k is the index of the prohibited operating zone, and Z
_{i}
is the number of prohibited operating zones (
Fig. 1
)
Cost function with Prohibited operating zones
 2.4 Ramp rate limit constraints
The generator constraints due to ramp rate limits of generating units are given as
P
As generation increases
As generation decreases
Therefore the generator power limit constraints can be modified as
From eqn. (9), the limits of minimum and maximum output powers of generating units are modified as
where P
_{i(t)}
is the output power of generating unit i in the time interval (t), P
_{i(t1)}
, is the output power of generating unit i in the previous time interval (t1), UR
_{i}
is the up ramp limit of generating unit i and DR
_{i}
is the down ramp limit of generating unit i.
The ramp rate limits of the generating units with all possible cases are shown in
Fig. 2
.
Ramp rate limits of generating units
3. Overview of EP and PSO
Fourdecade earlier EP was proposed for evolution of finite state machines, in order to solve a prediction task. Since then, several modifications, enhancements and implementations have been proposed and investigated. Mutation is often implemented by adding a random number or a vector from a certain distribution (e.g., a Gaussian distribution in the case of classical EP) to a parent. The degree of variation of Gaussian mutation is controlled by its standard deviation, which is also known as a ‘strategy parameter’ in an evolutionary search
[35]
. EP is near global stochastic optimization method starting from multiple points, which placed emphasis on the behavioral linkage between parents and their offspring rather than seeking to emulate specific genetic operators as observed in nature to find an optimal solution.
Particle Swarm Optimization (PSO) is a population based stochastic optimization technique which can be effectively used to solve the nonlinear and noncontinuous optimization problems. It inspired by social behavior of bird flocking or fish schooling. The PSO algorithm searches in parallel using a group of random particles similar to other AIbased optimization techniques.
Eberhart and Kennedy suggested a particle swarm optimization based on the analogy of swarm of bird and school of fish
[15]
. PSO is basically developed through simulation of bird flocking in two dimensional space. The position of each agent is represented by XY axis position, and also the velocity is expressed by Vx (velocity of X axis) and Vy (velocity of Y axis). Modification of the agent (particle) position is realized by the position and velocity information. Bird flocking optimizes a certain objective function. Each agent knows its best value so far (pbest) and its XY position. This information is the analogy of personal experiences of each agent. Moreover, each agent knows the best value so far in the group (gbest) among pbests. This information is the analogy of knowledge of how other agents around them have performed. The particles are drawn stochastically toward the position of present velocity of each particle, their prior best performance and the best previous performance of their neighbor
[16

17]
.
Each agent tries to modify its position using the following information:

1. The current position (x, y),

2. The current velocities (Vx, Vy),

3. The distance between the current position and pbest,

4. The distance between the current position and gbest.
This modification is represented by the concept of velocity. Velocity of each agent could be modified by the following Eq. (12)
Where ‘n’ is the population size, ‘m’ is the number of units and the ‘w’ be the inertia weight factor. Suitable selection of the inertia weight factors provides a balance between global and local explorations, thus requires fewer iteration on average to find a sufficiently optimal
[15]
. In general, the inertia weight
w
is set according to Eq. (13)solution
where,
Wmin and Wmax are the minimum and maximum weight factors respectively

Wmax = 0.9; Wmin =0.4

Iter – Current number of iterations

iter max – Maximum no of iterations (generations)

C1, C2– Acceleration constant, equal to 2

rand( ), Rand( ) – Random number value between 0 and 1

V(t)id– Velocity of agent i at iteration t

P(t)id– Current position of agent i at iteration t

pbest i – pbest of agent i

gbest – gbest of the group
Using the above equation, a certain velocity, which gradually gets closer to pbest and gbest, can be calculated. The current position can be modified by Eq. (14)
The first term of the righthand side of Eq. (12) is corresponding to the diversification in the search procedure. The second and third terms of that are corresponding to intensification in the search procedure. The PSO method has a wellbalanced mechanism to utilize the diversification and intensification in the search procedure efficiently.
Fig. 3
shows the concept of modification of a searching point by PSO.
Concept of modification of a searching point by PSO
where

Pt: Current searching point

Pt+1: Modified searching point

Vt: Current velocity

Vt+1: Modified velocity

V pbest : Velocity based on pbest

Vgbest : Velocity based on gbest
 3.1 Implementation of PSO for solving ED problem
The implementation of PSO method for solving ED problem is given as follows and the general flowchart of PSO is shown in
Fig. 4
.
General flowchart of PSO method
Step 1
. Generate an initial population of particles with random positions and velocities within the solution space
Step 2
. Calculate the value of the fitness function for each particle
Step 3
. To compare the fitness of each particle with each pbest. If the current solution is better than its pbest, then replace its pbest by the current solution.
Step 4
. Compare the fitness of all the particles with gbest. If the fitness of any particle is better than gbest, then replace gbest.
Step 5
. Modify the velocity and position of all particles according to Eqs. (12) & (14).
Step 6
. Repeat the steps 25 until a criterion is met.
4. Step by Step Development and Solution Methodology of the Proposed HPSO Method
Combining the special features of EP and PSO, the proposed HPSO has been developed, and the steps are given as follows.
 4.1 Step by step development of the HPSO method
Step 1
. Randomly generate the initial searching points of real power generation of generators and velocities within the allowable range. The current searching point is set to pbest for each agent. The best evaluated value of pbest is set to be gbest and gbest value is stored.
Step 2
. Modification of searching point of each agent is changed using Eqs. (12), (13) and (14) and the corresponding evaluation values are calculated.
Step 3
. If the evaluation value of each agent is better than the previous pbest, then the value is set to be pbest. If the best pbest is better than previous gbest, then the value is set to be gbest.
Step 4
. Modification of searching points using Gaussian mutation and the evaluation values are calculated.
Step 5
. If the evaluation value of each agent is better than the previous pbest, then the value is set to be pbest. If the best pbest is better than previous gbest, then the value is set to be gbest.
Step 6
. If the current iteration number reaches the predetermined maximum iteration number, then exit. Otherwise, go to step 2.
 4.2 Solution methodology of the proposed HPSO method to solve ED problem
The step by step procedure of the proposed HPSO method for solving ED problem is given below and the flow chart is shown in
Fig. 5
.
The flow chart of the proposed HPSO method
Step 1.
Specify the generation limits of each unit and calculate F
_{max}
and F
_{min}
. Randomly initialize the individuals of the population according to limits of each unit including velocity, search points and individual dimensions. This initial individual must be feasible candidate solution that satisfies the practical operating constraints. Initial velocity limits of each member in individual is
where,
Step 2.
For each Pi of the population use Bcoefficients loss formula given in Eq. (4) to calculate the transmission loss
Step 3.
Calculate the evaluation value of each individual Pi in the population using the Eq. (16)
where
F
_{max}
and F
_{min}
are the maximum and minimum generation cost among all individuals in the initial population respectively.
In order to limit the evaluation value of the each individual of the population with in a feasible range before estimating the evaluation value of an individual, the generation output power must satisfy the constraints
Step 4.
Compare each individual’s evaluation value with its pbest values. The best evaluation value among the pbest values is assigned as gbest value.
Step 5.
Modify the member velocity V of the each individual Pi using Eq. (12)
Step 6.
Check the velocity components constraint limits from the following conditions.
Step 7.
Modify the member position of each individual Pi using the Eq. (14)
must satisfy the constraints of prohibited operating zone and ramp rate limits.
Step 8.
If the evaluation value of each individual is better than the previous pbest value, then the current value is set to be pbest. If the best pbest is better than gbest, then the pbest is assigned as the gbest
Step 9.
created from each individual by Gaussian mutation
must satisfy the constraints of prohibited operating zones, ramprate limits and generator capacity limits
where,
f
_{i min}
Minimum cost among ‘n’ trial solutions, β scaling factor is equal to 0.001 and f
_{i}
 Value of the objective function associated with vector P
_{i}
.
Step 10.
If the evaluation value of each individual is better than the pbest value in step 8 then, the current value is set to be the pbest. If the best pbest among all particles is better than the gbest in step8, then, the value is set to be the gbest.
Step 11.
If the number of iterations reaches the maximum go to the step12. Otherwise go to the step 5.
Step 12.
The individual that generates the latest gbest is the optimal generation power of each unit with the minimum total generation cost.
5. Results and Discussion
To verify the feasibility of the proposed approach, four different test systems are considered such as three, six, fifteen and twenty units with ramp rate limits and prohibited operating zones constraints. Results of the proposed approach are compared with EP, conventional PSO and other methods, which are presented in the literatures. 100 trails runs were performed and observed the variations during the evolutionary process to reach convergence characteristics and optimal solutions. The Bloss coefficient matrix of power system network was employed to calculate the transmission line losses. The software was written in Mat Lab language and executed on the third generation Intel Core i3 processor based personal computer with 4 GB RAM. From the comparison of results, the proposed HPSO method is found to be better in solving the nonlinear ED problems.
Test System 1
A threeunit system
[36]
is considered. The system load demand is 300MW. The dimension of population is 100*3 and number of generations are 100. 100 trail runs are conducted, and the best solutions are shown in
Table 1
that satisfies the system constraints. The results of the proposed HPSO method are compared with EP, PSO, GA
[36]
and 2PNN
[37]
methods. From the comparison of the results, the fuel cost obtained by the proposed HPSO method is better than the other methods.
Fig. 6
shows the comparison of fuel costs for various methods in a three unit systems and
Fig. 7
shows the convergence nature of EP, PSO and HPSO methods. From the convergence property, it is evident that the proposed HPSO method has better convergence characteristics than EP and PSO method.
Results of three unit system with POZ and RRL
Results of three unit system with POZ and RRL
Comparison of fuel cost for 3 unit system
Convergence of EP, PSO and HPSO
Test system 2
The system contains six thermal units, 26 buses and 46 transmission lines
[15]
. The load demand is 1263MW. The losses are calculated using Bloss coefficient matrix. The dimension of the population is 100*6 and number of generations is taken as 100. 100 trial runs were conducted and the best solutions are shown in
Table 2
. The results obtained by the proposed method are compared with EP, conventional PSO, GA
[15]
, DSPSOTSA
[38]
, BBO
[39]
, HHS
[40]
, HIGA
[41]
and PSOGSA
[42]
methods. From the comparison of results, it clearly shows the proposed HPSO method gives minimum fuel cost than the other methods.
Fig. 8
shows the comparison of fuel cost for various methods in a six unit test system and
Fig. 9
shows the convergence nature of EP, conventional PSO and proposed HPSO methods.
Comparison of fuel cost for 6 unit system
Convergence of EP, PSO and HPSO
Results of six unit system with POZ and RRL
Results of six unit system with POZ and RRL
Test system 3
The input data of 15 unit test system are taken from reference
[15]
. The load demand of the system is 2630MW. The prohibited operating zones and ramprate limits are considered as the generator constraints. The losses are calculated using Bloss coefficient matrix. The dimension of the population is 100*15 and number of generations is taken as 100. The results obtained by the proposed method is compared with EP, PSO, GA
[15]
, PSOMSAF
[43]
, GAAFI
[44]
and TVACEPSO
[45]
methods and are shown in
Table 3
. From the comparison of results, it is observed that the proposed HPSO method gives minimum fuel cost than the other methods.
Fig. 10
shows the fuel cost comparison for various methods in a fifteen unit test system and
Fig. 11
shows the convergence nature EP, PSO and proposed HPSO methods.
Fuel cost comparison for 15 unit system
Convergence of EP, PSO and HPSO
Results of fifteen unit system with POZ and RRL
Results of fifteen unit system with POZ and RRL
Test system 4
The input data for 20 unit test system is taken from
[46]
.The system load demand is 2500 MW. In this test system, the transmission losses, POZ and ramp rate limit constraints are considered. The dimension of the population is 100*20 and the number of generations are100.The results obtained by the proposed method is compared with EP, PSO, Lambdaiteration method
[46]
, Hopfield neural network method
[46]
, BBO
[47]
and EBBO
[48]
methods and are shown in
Table 4
. On comparison of the results, it is evident that the proposed method can provide significant cost saving than other methods.
Results of twenty unit system with POZ and RRL
Results of twenty unit system with POZ and RRL
Fig. 12
shows the fuel cost comparison for various methods for a 20 unit test system and
Fig. 13
shows the convergence nature EP, PSO and proposed HPSO methods. It’s evident from the
Figs. 7
,
9
,
11
,
13
, the proposed HPSO method is free from the shortcoming of premature convergence exhibited by the EP and PSO methods.
Fuel cost comparison for 20 unit system
Convergence of EP, PSO and HPSO
6. Conclusion
In this paper, EP, conventional PSO, and proposed HPSO are applied successfully to solve the nonlinear economic dispatch problems. The proposed HPSO method has been proved to have superior features in terms of achieving better optimal solutions for reducing the fuel cost of the generating units and improving the convergence characteristics. Nonlinear characteristics of the generators such as prohibited operating zones and ramprate limits constraints are considered for the selected test systems. The result obtained by the proposed HPSO method is compared with EP, conventional PSO and other methods reported in recent literatures. The comparative study was done based on the optimum fuel cost. From this study, it can be concluded that the proposed HPSO method can be an alternative approach for finding a better solution for the non linear economic dispatch problems.
BIO
S. Prabakaran was born in 1968. He received A.M.I.E (Electrical Engineering) degree from The Institution of Engineers (INDIA) in 1993 and M.E degree in power system from Annamalai University, Chidambaram, India in 1999. He is currently doing Ph.D in Power System at College of Engineering, Guindy, Anna University, Chennai, India. He has published research papers in International journals and conferences. His research interests are Power system optimization, Operation and Control and Deregulated power supply.
V. Senthilkumar was born in 1971. He received B.Tech degree (Electrical Engineering) in 1995 from Regional Engineering College, Hamirpur, India and M.E degree in power system from Annamalai University, Chidambaram, India in 1998. He received Ph.D at college of Engineering, Guindy, Anna University, Chennai, India in 2009. He is currently working as Associate Professor in Electrical and Electronics Engineering department, at Anna University, Chennai, India. He has more than 18 years experience in teaching, research and published many research papers in leading International journals and conferences. His research interests include AI technique to Power system optimization problems and Operational planning and control in restructured power system.
G. Baskar was born in 1967. He received A.M.I.E (Electrical Engineering) degree from The Institution of Engineers (INDIA) in 1993, M.E and Ph.D degrees in power system at College of Engineering, Guindy Anna University, Chennai, India in 2001 and 2008 respectively. He has more than 21 years of experience in teaching, research and published many research papers in leading International journals and conferences. His research interests include AI technique to Power system optimization problems and Operational planning and control in restructured power system.
Chowdhury B.H.
,
Rahman S.
1990
“A review of recent advances in economic dispatch”
IEEE Trans. Power Systems
5
(4)
1248 
1259
Wood A. J.
,
Wollenberg B.F.
1996
“Power generation operation and control”
2nd edition
Wiley
New York
29 
32
2009
“Application of Soft Computing methods for Economic Dispatch in Power Systems”
International Journal of Electrical and Electronics Engineering
3
(9)
538 
543
Liang Z.X.
,
Glover J.D.
1992
“A Zoom feature for a Dynamic Programming solution to economic dispatch including transmission losses’”
IEEE Transactions on Power Systems
7
(2)
544 
549
DOI : 10.1109/59.141757
Selvakumar A. J.
,
Thanushkodi K.
2007
“A new particle swarm optimization solution to nonconvex economic dispatch problems”
IEEE Trans. Power Systems
22
(1)
42 
51
Wong K.P.
,
Fung C.C.
1993
“Simulated annealing based economic dispatch algorithm, “
IEE proceedings Generation, Transmission and Distribution
140
(6)
509 
515
DOI : 10.1049/ipc.1993.0074
Desilva I. N.
,
Nepomuceno L.
,
Basdo T. M.
2002
“Designing a modified Hopfield Network to solve an ED problem with nonlinear cost function”
proceeding Int. conf on Neural networks
2
1160 
1165
Baskar G.
,
Kumarappan N.
,
Mohan M. R.
2003
“Optimal Dispatch using Improved Lambda based Genetic algorithm suitable for utility system”
Int. journal on Electric Power components and systems
31
627 
638
DOI : 10.1080/15325000390203647
Yang H.T.
,
Yang P.C.
,
Huang C.L.
1996
“Evolutionary programming based economic dispatch for units with nonsmooth fuel cost functions”
IEEE Trans. on Power Systems
11
112 
118
DOI : 10.1109/59.485992
Wong K. P.
,
Yuryevich J.
1998
“Evolutionary program based algorithm for environmentally constrained economic Dispatch ”
IEEE Trans. Power System
13
301 
306
DOI : 10.1109/59.667339
Yao X.
,
Liu Y.
,
Lin G.
1999
“Evolutionary Programming made faster”
IEEE Trans. Evol. Comput
3
82 
102
DOI : 10.1109/4235.771163
Jayabharathi T.
,
Sadasivam G.
,
Rama chandran V.
1999
“Evolutionary Programming based economic dispatch of generators with prohibited operating zones”
Electric Power System Research
52
261 
266
DOI : 10.1016/S03787796(99)000255
Jayabharathi T.
,
Jayaprakash K.
,
Jeyakumar D. N.
,
Raghunathan T.
2005
“Evolutionary Programming Techniques for different kinds of Economic dispatch problems,”
Electric Power Systems Research
73
169 
176
DOI : 10.1016/j.epsr.2004.08.001
Lin W. M.
,
Cheng F.S.
,
Tsay M. T.
2002
“An improved Tabu search for economic dispatch with multiple minima,”
IEEE Trans. on Power Systems
17
(1)
108 
112
DOI : 10.1109/59.982200
Gaing ZweLee
2003
“Particle Swarm Optimization to solving the Economic Dispatch considering the generator constraints”
IEEE Trans. on Power Systems
18
(3)
1187 
1195
Park J. B.
,
Lee K.
,
Shin J.
,
Lee K. Y.
2005
“A particle swarm optimization for economic dispatch with nonsmooth cost functions”
IEEE Trans. on Power Systems
20
(1)
34 
42
DOI : 10.1109/TPWRS.2004.831275
Jeyakumar D. N.
,
Jayabarathi T.
,
Raghunathan T.
2006
“A Particle Swarm Optimization for various types of economic dispatch problems,”
International journalon Electric Power and Energy System
28
(1)
36 
42
DOI : 10.1016/j.ijepes.2005.09.004
Alrashidi M.R.
,
EI Hawary M. E.
2006
“A Survey of Particle Swarm Optimization applications in power system operations,”
Electric Power components and systems
34
(12)
1349 
1357
DOI : 10.1080/15325000600748871
Chen ChunLung
,
Jan RongMow
,
Lee TsungYing
,
Chen ChengHsiung
2011
“A novel particle swarm optimization Algorithm solution of Economic Dispatch with Valve Point Loading”
Journal of Marine Science and Technology
19
(1)
43 
51
Geem Zong woo
2013
“Economic Dispatch Using ParameterSetting_Free Harmony Search”
journal of Applied Mathematics
2013
1 
5
Subbaraj P.
,
Rengaraj R.
,
Salivahanan S.
2009
“Enhancement of Combined heat and power economic dispatch using Self adaptive real coded genetic algorithm”
Applied Energy
86
915 
921
DOI : 10.1016/j.apenergy.2008.10.002
MohammadiIvatloo B.
,
Rabiee A.
,
Soroudi A.
,
Ehsan M.
2012
“Iteration PSO with time varying acceleration coefficient for solving non convex economic dispatch problems”
Electrical Power and Energy Systems
42
508 
516
DOI : 10.1016/j.ijepes.2012.04.060
Shi Y.H.
,
Eberhart R.C.
2001
“Fuzzy Adaptive particle swarm optimization”
Proc. of the IEEE Congress on Evolutionary Computation
Seoul Korea
1
101 
106
Shi Y.H.
,
Eberhart R.C.
1998
“A modified particle swarm optimizer”
Proc. of the IEEE Congress on Evolutionary Computation
IEEE Service Center,USA
69 
73
Zhang Y.L.
,
Ma L.H.
,
Zhang L.Y.
,
Qian J.X.
2003
“On the Convergence Analysis and Parameter Selection in Particle Swarm Optimization”
Proc. Int. Conf. on Machine learning and Cybernetics
Zhejiang University, Hangzhou, China
1802 
1807
Eberhart R.C.
,
Shi Y.H.
2001
“Tracking and optimizing dynamic systems with particle swarms”
Proc. of the IEEE Congress on Evolutionary Computation
San Francisco, Cailf, USA
94 
100
Zhang L. P.
,
Yu H. J.
,
Chen D. Z.
,
Hu S. X.
2004
“Analysis and improvement of particle swarm optimization algorithm”
Information and Control
33
513 
517
Wang S.K.
,
Chiou J.P.
,
Liu C.W.
2007
“Nonsmooth / nonconvex economic dispatch by a novel hybrid differential evolution algorithm”
IET Gen., Transm., Distribution
1
(5)
793 
803
DOI : 10.1049/ietgtd:20070183
Chiou J. P.
2007
“Variable scaling hybrid differential evolution for large scale economic dispatch problems”
Elect. Power System Research
77
(1)
212 
218
DOI : 10.1016/j.epsr.2006.02.013
Chiang CL.
2007
“Genetic based algorithm for power economic load dispatch”
IET Gen., Transm., Distribution
1
(2)
261 
269
DOI : 10.1049/ietgtd:20060130
Mariani V.C.
,
Coelho L.D.S.
2008
“Particle swarm approach based on quantum mechanics and harmonic oscillator potential well for economic load dispatch with valvepoint effects”
Energy Converse. Management
49
(11)
3080 
3085
DOI : 10.1016/j.enconman.2008.06.009
Chaturvedi K.T.
,
Pandit M.
,
Srivastava L.
2008
“SelfOrganizing hierarchical particle swarm Optimization for nonconvex economic dispatch”
IEEE Trans on Power System
23
(3)
1079 
1087
Pandi B.K.
,
Panigrahi V.R.
2008
“Bacterial foraging optimization; NelderMead hybrid algorithm for economic load dispatch”
IET Gen., Transm., Distribution
2
(4)
556 
565
DOI : 10.1049/ietgtd:20070422
Sinha Nidul
,
Chakrabarti R.
,
Chattopadhyay P. K.
2003
“Evolutionary Programming Techniques for Economic Load Dispatch”
IEEE Trans. on Evolutionary Computation
7
(1)
83 
94
DOI : 10.1109/TEVC.2002.806788
Chen PH.
,
Chang HC.
1995
“Large scale economic dispatch by genetic algorithm”
IEEE Trans.on Power Systems
10
(4)
1919 
1926
DOI : 10.1109/59.476058
Naresh R.
,
Dubey J.
,
Sharma J.
2004
“Twophase neural network based modeling frame work of constrained economic load dispatch”
IEE proce. Generation, Transmission, Distribution
151
(3)
373 
378
DOI : 10.1049/ipgtd:20040381
Jiriwibhakorn S.
2010
“DSPSOTSA for economic dispatch problem with non smooth and non continuous cost functions”
Energy Conversation Management
51
365 
375
DOI : 10.1016/j.enconman.2009.09.034
Bhattacharya A.
,
Chattopadhyay P. K.
2010
“Bio geography Based Optimization for different economic load dispatch problems”
IEEE Trans. on power Systems
25
(2)
1064 
1077
DOI : 10.1109/TPWRS.2009.2034525
Pandi V. R.
,
Panigrahi B. K.
,
Bansal R. C.
,
Das S.
,
Mohapatra A.
2011
“Economic Load Dispatch using Hybrid Swarm Intelligence Based Harmony search Algorithm”
Electric power components and systems
39
(8)
751 
767
DOI : 10.1080/15325008.2010.541411
Mahmood Hosseini Mir.
,
ghorbani Hamidreza
,
Rabii A.
,
Anvari Sh.
2012
“A novel Heuristic Algorithm for solving Nonconvex Economic Load Dispatch problem with Nonsmooth cost function”
Journal of Basic and Applied Scientific Research
2
(2)
1130 
1135
Dubey Hari Mohan
,
Pandit Manjaree
,
Panigrahi Mugdha Udgir B. K.
2013
“Economic Load Dispatch by Hybrid Swarm Intelligence Based Gravitational Search Algorithm”
I. J Intelligent Systems and Applications
08
21 
32
Subbaraj P.
,
Rengaraj R.
,
Salivahanan S.
,
Senthilkumar T.R.
2010
“Parallel Particle Swarm Optimization with modified stochastic acceleration factors for solving large scale economic dispatch problem”
International journal of Electrical Power and Energy Systems
32
1014 
1023
DOI : 10.1016/j.ijepes.2010.02.003
Ciornei I.
,
Kyriakides E.
2012
“A GAAPI solution for the Economic Dispatch of Generation in Power System Operation”
IEEE Transaction on Power System
27
233 
242
DOI : 10.1109/TPWRS.2011.2168833
ABDULLAH Mohd Noor
,
RAHIM Nasrudin Abd
,
BAKAR Abd Halim Abu
,
MOKHLIS Hazlie
,
ILLIAS Hazlee Azhil
,
JAMIAN Jasrul Jamani
2013
“Efficient Evolutionary Particle Swarm Optimization Approach for Non convex Economic Load Dispatch Problem”
PRZEGLAD ELEKTROTECHNICZNY
R 89
(NR 2a)
139 
143
Su ChingTzong
,
Lin Chen Tung
2000
“New Approach with a Hopfield Modelling Frame work to Economic Dispatch”
IEEE trans on power system
15
(2)
541 
545
DOI : 10.1109/59.867138
Bhattacharya A.
2010
“BiogeographyBased Optimization for different economic load dispatch problems”
IEEE Trans on power systems
25
(2)
1064 
1077
DOI : 10.1109/TPWRS.2009.2034525
Vanitha M.
,
Thanuskodi K.
2012
“An Effective Biogeography Based Optimization Algorithm to solve Economic Load Dispatch Problems”
Journal of computer sciences
8
(2)
1482 
1486